Let $M$ be a $W$ module on which $C$ acts trivially. Then by Hochschild-serre gives a spectral sequence $$0 \rightarrow H^{1}(G,M) \rightarrow H^{1}(W,M) \rightarrow H^{1}(C,M)^{G} \overset{\tau}\rightarrow H^{2}(G,M) $$ where $\tau$ is the transgression map. Now he goes on to prove that in this case $\tau(\alpha) = - \alpha \cup u$ (cup-product) for some $\alpha \in H^{0}(G,Hom(C,M))$.

Now consider the LHS for group homology, with $C,W,G,M$ as above ($C$ acting trivially on $M$)

1 Answer
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The homology analogue of the result stated by Milne is that the transgression in the Lyndon-Hochschild-Serre spectral sequence for group homology
$$
H_2(G;M)\to H_0(G;H_1(C;M)) = H_1(C;M)_G
$$
is given (up to a sign) by cap product with the extension class $H^2(G;C)$. This makes sense since the cap product is a map
$$
H^2(G;C)\otimes H_2(G;M)\to H_0(G;C\otimes M)=H_0(G;H_1(C; M)),
$$
the final equality since $C$ is abelian.

This is supposedly well-known, see page 3 of this paper. I don't know a reference to an explicit proof, though.